Question

When the play button is pressed, a cd accelerates uniformly from rest to 480 rev/min in 2.0 revolutions. if the cd has a radius of 6.0 cm and a mass of 20 g , what is the torque exerted on it

Express your answer using two significant figures.
=_______N*m

Answer 1

To find the torque exerted on the CD, we calculate the angular acceleration using the formula for change in angular velocity and change in time. We find the number of revolutions using the formula for change in angular velocity, and then calculate the torque using the formula for moment of inertia and angular acceleration.

Step-by-step explanation:

To find the torque exerted on the CD, we need to calculate the moment of inertia of the CD and multiply it by its angular acceleration. The moment of inertia of a solid cylinder can be calculated using the formula I = (1/2) * m * r^2, where I is the moment of inertia, m is the mass, and r is the radius.

Since the CD is accelerated uniformly from rest, we can use the formula for angular acceleration, α = Δω / Δt, where α is the angular acceleration, Δω is the change in angular velocity, and Δt is the change in time. The change in angular velocity can be calculated using the formula Δω = ωf - ωi, where ωf is the final angular velocity and ωi is the initial angular velocity.

(a) To calculate the angular acceleration, we need to convert the change in angular velocity to rad/s using the formula Δω = (2π * ΔN) / 60, where ΔN is the change in revolutions per minute. Plugging in the values, we get Δω = (2π * 480) / 60 = 32π rad/s. Then, we can calculate the angular acceleration using the formula α = Δω / Δt. Plugging in the values, we get α = (32π rad/s) / (2.0 rev) = 16π rad/s².

(b) To calculate the number of revolutions, we can use the formula ΔN = ωf * Δt, where ΔN is the change in revolutions, ωf is the final angular velocity in rev/s, and Δt is the change in time. Plugging in the values, we get ΔN = (480 rev/min * 2.0 min) / (60 min/s) = 16 rev.

Now that we have the angular acceleration and the number of revolutions, we can calculate the torque exerted on the CD using the formula τ = I * α, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. Plugging in the values, we get τ = (1/2) * (0.02 kg) * (0.06 m)^2 * 16π rad/s² = 0.000288π N*m ≈ 0.00091 N*m (to two significant figures).